Geometry and Mensuration

The area of a square with side length s is s². The diagonal of this square is s√2, so the square drawn on the diagonal has area (s√2)² = 2s². Therefore, the ratio of the area of the original square to the square on its diagonal is s² : 2s² = 1 : 2.

The perimeter of a square is 4 times its side length. The side length of a square can be found from its diagonal using the relation side = diagonal / √2. If one square has a diagonal 3 times the length of the other, then its side length is also 3 times the other (since side is proportional to diagonal). Therefore, the ratio of their perimeters is also 3:1. Among the given options, this corresponds to option D.

The area of a circle is proportional to the square of its radius. Given the radii ratio of 3:5, the ratio of their areas is 3²:5² = 9:25. The difference in area is 25 - 9 = 16 parts. To find what percent less the area of the first field is compared to the second, calculate (16/25) × 100% = 64%. Therefore, the first field's area is 64% less than the second field's area. This matches option D.

When the diameter of a circle is increased by 100%, it doubles. Since the area of a circle is proportional to the square of the diameter, the area increases by (2)^2 = 4 times, which is a 400% increase.

The area of the verandah is the difference between the area of the hall plus verandah and the hall alone. The total dimensions including the verandah are (20 + 2*2.5) m by (15 + 2*2.5) m = 25 m by 20 m, so area = 500 m². The hall area is 20 m × 15 m = 300 m². The verandah area is 500 - 300 = 200 m². At Rs 3.50 per m², the cost is 200 × 3.50 = Rs 700.

The problem involves a rectangular field where the diagonal length is 20 m and the length is 16 m. Using the Pythagorean theorem for a right triangle formed by the length, breadth, and diagonal, we have: breadth = √(diagonal² - length²) = √(20² - 16²) = √(400 - 256) = √144 = 12 m. Therefore, the breadth of the field is 12 m, which corresponds to option B.

The volume of earth dug out is 6 × 3 × 2.5 = 45 m³. The total area of the field is 20 × 14 = 280 m². The pit occupies an area of 6 × 3 = 18 m², so the remaining area is 280 - 18 = 262 m². The earth is spread uniformly over this remaining area, so the increase in level = volume / area = 45 / 262 ≈ 0.1717 m = 17.17 cm. The closest option to this value is C (17.18 cm).

The total dimensions including the path are (75 + 2*2.5) = 80 m and (55 + 2*2.5) = 60 m. The total area including the path is 80 * 60 = 4800 sq m. The area of the field alone is 75 * 55 = 4125 sq m. Therefore, the area of the path is 4800 - 4125 = 675 sq m. At Rs.2 per sq m, the cost is 675 * 2 = Rs.1350.

Given one side of the rectangle is 4 m and the diagonal is 5 m, we can use the Pythagorean theorem to find the other side: let the other side be x, then 4² + x² = 5², so 16 + x² = 25, which gives x² = 9 and x = 3 m. The area is length × width = 4 × 3 = 12 sq m.

Using the Pythagorean theorem, the height of the right triangle is calculated as √(10² - 8²) = √(100 - 64) = √36 = 6. The area of the right triangle is (1/2) × base × height = (1/2) × 8 × 6 = 24. Therefore, the correct answer is option D (24).

The original area is 159 cm² and the length is 15 cm, so the width is 159 ÷ 15 = 10.6 cm. The new area is increased to 1 1/3 times the original area, which is (4/3) × 159 = 212 cm². Since only the length is increased, the new length is 212 ÷ 10.6 ≈ 20 cm. The new perimeter is 2 × (length + width) = 2 × (20 + 10.6) = 61.2 cm, which is closest to option B (60 cm). Therefore, the correct answer is B, not C.

The area of the square is x². The area of the triangle is (1/2) × base × altitude = (1/2) × x × altitude. Setting these equal: x² = (1/2) × x × altitude, solving for altitude gives altitude = 2x.

The given sides 1m, 2m, and 3m do not satisfy the triangle inequality (1 + 2 = 3), so these lengths cannot form a triangle. Therefore, the area is 0 square meters.

The altitude of an equilateral triangle with side length s is given by (s√3)/2. Here, s = 2√3 cm, so altitude = (2√3 × √3)/2 = (2 × 3)/2 = 3 cm.

Given the circumference C = 352 meters, we use the formula C = 2πr to find the radius r = C/(2π) = 352/(2×3.14) ≈ 56 meters. Then, the area A = πr² ≈ 3.14 × 56² = 3.14 × 3136 ≈ 9856 square meters, which matches option A.

The area of a circle is proportional to the square of its radius. If the radius is reduced by 50%, the new radius is 0.5 times the original. Therefore, the new area is (0.5)^2 = 0.25 times the original area, meaning the area is reduced by 75%. Hence, the correct answer is 75%.

The perimeter of a semicircle is given by the formula P = πr + 2r. Given P = 144 cm, solving 144 = r(π + 2) gives r ≈ 30 cm.

Let the length and width of the carpet be L and W. Given area = L × W = 120 and perimeter = 2(L + W) = 46, so L + W = 23. Solving these, we get L and W as roots of x^2 - 23x + 120 = 0, which are 15 and 8. The diagonal length = √(L² + W²) = √(15² + 8²) = √(225 + 64) = √289 = 17 m.

The area of a quadrilateral given one diagonal and its offsets (perpendicular distances from the diagonal to the opposite vertices) is calculated as half the product of the diagonal and the sum of the offsets: Area = 1/2 × diagonal × (offset1 + offset2) = 1/2 × 20 × (9 + 6) = 150 cm².

The area of a circle is calculated using the formula A = πr². With a radius of 7 m, the area is π × 7² = π × 49 ≈ 154 sq m, which corresponds to option B.

The original figure is a square with side length 's', so its area is s². When the length is increased by 40%, the new length becomes 1.4s. When the breadth is increased by 30%, the new breadth becomes 1.3s. The new area of the rectangle is therefore 1.4s × 1.3s = 1.82s². This means the area has increased by (1.82 - 1) × 100% = 82%. Hence, the area of the resulting rectangle exceeds the area of the original square by 82%, which corresponds to option C.

Let the length be 5x and breadth be 4x. Given breadth is 20 meters less than length, so 5x - 4x = 20, which gives x = 20. Therefore, length = 100 m and breadth = 80 m. The perimeter is 2(length + breadth) = 2(100 + 80) = 360 m. However, this contradicts the calculation; rechecking: 5x - 4x = 20 => x=20, length=100, breadth=80, perimeter=2(100+80)=360 m. The correct perimeter is 360 m, so option C is correct.

When the radius of a circle increases by 1%, the new radius is 1.01 times the original. The area depends on the square of the radius, so the area increases by (1.01)^2 - 1 = 0.0201 or 2.01%. Therefore, the area increases by approximately 2.01%.

Let the breadth be x meters. Then the length is 4.5x. The area is length × breadth = 4.5x × x = 4.5x² = 200. Solving for x² gives x² = 200/4.5 ≈ 44.44, so x ≈ 6.67 meters. The length is 4.5 × 6.67 ≈ 30 meters. However, this contradicts the calculation; rechecking: 4.5x² = 200 → x² = 200/4.5 ≈ 44.44 → x ≈ 6.67. Length = 4.5 × 6.67 ≈ 30 meters. So length is approximately 30 meters, which corresponds to option B. Therefore, the original correct_answer B is correct. The initial explanation was missing, so it has been added to clarify the solution.

In a rhombus, the diagonals are perpendicular bisectors. Given one diagonal is 20 m, half of it is 10 m. Using the Pythagorean theorem with side 26 m, the half of the other diagonal is √(26² - 10²) = √(676 - 100) = √576 = 24 m. So, the other diagonal is 48 m. The area is (1/2) × 20 × 48 = 480 sq m. However, the calculation shows 480 sq m, which matches option D, not B. Rechecking: half diagonal 1 = 10, side = 26, half diagonal 2 = √(26² - 10²) = √(676 - 100) = √576 = 24, so full diagonal 2 = 48. Area = (1/2) × 20 × 48 = 480 sq m. So the correct answer is D, not B. The original answer D is correct. The explanation was missing and is now added. The topic is geometry, difficulty medium, and tags added accordingly.